“What am I supposed to do if I ever have to meet the person my wife cheated on me with? How should I treat him and let him know I disapprove of his ‘taking my wife from me’ and that he is not, and never will be my two children’s father, but that I am ‘not out for blood’ in vengeance against him?”

Offer grace
That your wife cheated reflects thoroughly on your wife’s character for it takes two to tangle (have conflict) but only one to cheat. That he too is a cheat (she did not cheat alone) means that it is up to you, for the children’s sake, to take the high road. Therefore I challenge you to do your part to reduce your children’s anxieties (they are not immune to the destructiveness of their mother’s actions) by NOT making this man or your ex-wife your enemy.
The situation you face goes to the heart of what it means to be a man. Can you rise up and do what is right and good and healthy for your children when others are choosing not to do so? Call me. I’d love to talk – these are “primordial” issues and ought not be faced alone.